The Modified Korteweg-de Vries Equation on the Half-Line with a Sine-Wave as Dirichlet Datum

Boundary value problems for integrable nonlinear evolution PDEs, like the modified KdV equation, formulated on the half-line can be analyzed by the so-called unified transform method. For the modified KdV equation, this method yields the solution in terms of the solution of a matrix Riemann-Hilbert problem uniquely determined in terms of the initial datum q(x,0), as well as of the boundary values {q(0,t), q(x)(0,t), q(xx)(0,t)}. For the Dirichlet problem, it is necessary to characterize the unknown boundary values q(x)(0,t) and q(xx)(0,t) in terms of the given data q(x,0) and q(0,t). It is shown here that in the particular case of a vanishing initial datum and of a sine wave as Dirichlet datum, q(x)(0, t) and q(xx)(0,t) can be computed explicitly at least up to third order in a perturbative expansion and that at least up to this order, these functions are asymptotically periodic for large t.